Consider the i.i.d. Gaussian variables $$z_i \sim N(\mu,\sigma)$$, where $$\mu=0$$ and $$\sigma=1$$. The random variables from a linear combination of $$z_i$$, e.g.

\begin{aligned} x_1 &= a_{11} z_1 + a_{12} z_2 + \cdots + a_{1n} z_n \\ x_2 &= a_{21} z_1 + a_{22} z_2 + \cdots + a_{2n} z_n \end{aligned}

are correlated. Their covariance is given by

\begin{aligned} cov(x_1, x_2) &= E[(x_1 - \bar{x}_1)(x_2 - \bar{x}_2)] \\ &= E[(a_{11} z_1 + a_{12} z_2 + \cdots + a_{1n} z_n)(a_{21} z_1 + a_{22} z_2 + \cdots + a_{2n} z_n)] \\ &= a_{11}a_{21} + a_{12}a_{22} + \cdots + a_{1n}a_{2n} \end{aligned}

as the covariance between two independent standard normal variables is zero.

Consider a $$k$$-vector $$\vec{z}$$ of independent standard normal variables $$z_i$$, and define the $$n$$-vector of random variables $$\vec{x}$$ as the linear combination of the $$k$$ variables in $$\vec{z}$$ and a $$n$$-vector of constants $$\vec{\mu}$$:

$\vec{x} = \mathbf{A}\vec{z} + \vec{\mu},$

then the covariance matrix of $$\vec{x}$$ is $$\mathbf{\Sigma} = \mathbf{AA}^T$$ and the mean is $$\vec{\mu}$$.

The p.d.f. of a univariate Gaussian variable is

$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp(-\frac{(x-\mu)^2}{2\sigma^2})$

while the p.d.f. of a multivariate Gaussian variable ($$n$$-vector) is

$f(\vec{x}) = \frac{1}{\sqrt{(2\pi)^n|\mathbf{\Sigma}|}} \exp(-\frac{1}{2}(\vec{x}-\vec{\mu})^T\mathbf{\Sigma}^{-1}(\vec{x}-\vec{\mu}))$